Finite Group Databasebeta

← #10956135

Subgroups — #10956135

Order 768.

3106 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5574C211yesmaximal, characteristic
2C2×#5574C211yesmaximal, characteristic
3C2×C4×#239C211yesmaximal, characteristic
4C3×Q8×C4⋊C4C211yesmaximal, characteristic, complemented
5#82461C211yesmaximal, characteristic, complemented
6#82460C211yesmaximal, characteristic, complemented
7Q8×Dic3⋊C4C211yesmaximal, characteristic, complemented
8#76203C213yesmaximal, complemented
9#76202C213yesmaximal, complemented
10#76195C213yesmaximal, complemented
11D3×#389C213yesmaximal, complemented
12#74360C213yesmaximal, complemented
13#74359C213yesmaximal, complemented
14#73770C213yesmaximal, complemented
15#73755C213yesmaximal, complemented
16#34250—31nomaximal, Sylow
17C22×#239C2211yescharacteristic
18C2×C3×C4×Q8C2211yescharacteristic
19C2×C3×C4×Q8C2211yescharacteristic
20C2×C3×C4×Q8C2211yescharacteristic
21C2×Q8×Dic3C2211yescharacteristic
22C2×Q8×Dic3C2211yescharacteristic
23C2×Q8×Dic3C2211yescharacteristic
24#5574—28no—
25#5574C4116yescomplemented
26C4×#239—28no—
27C2×#893C2213yes—
28C2×#893C2213yes—
29C2×#893C2213yes—
30C2×#892C2213yes—
31C2×D3×C4⋊C4C2213yes—
32C2×D3×C4⋊C4C2216yes—
33C2×D3×C4⋊C4C2213yes—
34C2×C4×D12C2213yes—
35C2×C4×D12C2213yes—
36C2×C4×D12C2213yes—
37C2×#872C2213yes—
38C2×#872C2213yes—
39C2×C42×D3C2213yes—
40C3×#389C2213yes—
41C3×#389C2213yes—
42C3×#384C2213yes—
43C3×C4×C4⋊C4C2213yes—
44#4989C2213yes—
45#4989C2213yes—
46#4989C2216yes—
47#4989C2216yes—
48#4987C2216yes—
49#4984C2213yes—
50#4984C2213yes—
51#4977C2213yes—
52#4976C2211yescharacteristic
53#4976C2213yes—
54#4976C2216yes—
55#4975C2213yes—
56#4972C2213yes—
57#4972C2213yes—
58#4936C2213yes—
59C4×C4⋊Dic3C2213yes—
60C4×Dic3⋊C4C2213yes—
61C4×Dic3⋊C4C2213yes—
62#4928C2213yes—
63#4928C2213yes—
64#4668C2213yes—
65#4668C2213yes—
66#4667C2216yes—
67D3×#95C2216yes—
68#4647C2213yes—
69#4647C2213yes—
70C2×#520—31no—
71C2×#520—31no—
72C2×C4×#41—31no—
73#2351—31no—
74#2351—31no—
75Q8×C4⋊C4—31no—
76Q8×C4⋊C4D311yescharacteristic, complemented
77#2349—33no—
78#2349—33no—
79D4×C4⋊C4—33no—
80D4×C4⋊C4—33no—
81#2317—33no—
82#2316—33no—
83#2296—33no—
84C2×#389—33no—
85C22×C3×Q8C2311yescharacteristic
86C2×#239C2×C418yes—
87C2×#239D418yes—
88C2×#239Q818yes—
89C22×D12C2313yes—
90C22×C4×D3C2313yes—
91C22×C4×D3C2313yes—
92C22×C4×D3C2316yes—
93C22×C4×D3C2316yes—
94C22×C4×D3C2313yes—
95C3×C4×Q8—24no—
96C3×C4×Q8—24no—
97C3×C4×Q8C2×C418yescomplemented
98C2×C3×C4⋊C4C2313yes—
99C2×C3×C4⋊C4C2313yes—
100C2×C3×C4⋊C4C2311yescharacteristic
101C2×C3×C4⋊C4C2316yes—
102C2×C3×C4⋊C4C2313yes—
103C2×C3×C4⋊C4C2313yes—
104C2×C3×C42C2313yes—
105C2×C3×C42C2313yes—
106C2×C3×C42C2313yes—
107Q8×Dic3—24no—
108Q8×Dic3—24no—
109Q8×Dic3C2×C418yescomplemented
110C2×(C2×D3)⋊C4C2313yes—
111C2×(C2×D3)⋊C4C2316yes—
112C2×(C2×D3)⋊C4C2313yes—
113C2×(C2×D3)⋊C4C2313yes—
114C2×(C2×D3)⋊C4C2313yes—
115C2×C4⋊Dic3C2313yes—
116C2×C4⋊Dic3C2313yes—
117C2×C4⋊Dic3C2313yes—
118C2×C4⋊Dic3C2313yes—
119C2×C4⋊Dic3C2313yes—
120C2×Dic3⋊C4C2311yescharacteristic
121C2×Dic3⋊C4C2313yes—
122C2×Dic3⋊C4C2313yes—
123C2×Dic3⋊C4C2311yescharacteristic
124C2×Dic3⋊C4C2313yes—
125C2×Dic3⋊C4C2316yes—
126C2×Dic3⋊C4C2313yes—
127C2×Dic3⋊C4C2313yes—
128C2×Dic3⋊C4C2311yescharacteristic
129C2×C4×Dic3C2311yescharacteristic
130C2×C4×Dic3C2313yes—
131C2×C4×Dic3C2313yes—
132C2×C4×Dic3C2313yes—
133C2×C4×Dic3C2313yes—
134#893—212no—
135#893—212no—
136#893C2×C4124yescomplemented
137#892—212no—
138D3×C4⋊C4—212no—
139D3×C4⋊C4—412no—
140D3×C4⋊C4C2×C4124yescomplemented
141C4×D12—212no—
142C4×D12—212no—
143C4×D12C2×C4124yescomplemented
144#872—212no—
145#872C2×C4124yescomplemented
146C42×D3—212no—
147C3×#95C2313yes—
148C3×#95C2313yes—
149C3⋊#95C2313yes—
150C3⋊#95C2313yes—
151C3⋊#95C2316yes—
152C3⋊#95C2316yes—
153C3⋊#95C2311yescharacteristic
154C3⋊#95C2313yes—
155C3⋊#95C2311yescharacteristic
156C3⋊#95C2313yes—
157C22×#41—31no—
158#520—316no—
159#520—68no—
160C4×#41—68no—
161C2×C4×Q8—31no—
162C2×C4×Q8—31no—
163C2×C4×Q8—31no—
164C2×C4×Q8C2×D311yescharacteristic
165C2×C4×Q8C2×D311yescharacteristic
166C2×C4×Q8C2×D311yescharacteristic
167C2×C4×D4—33no—
168C2×C4×D4—33no—
169C2×C4×D4—33no—
170C2×C4×D4—33no—
171C2×C4×D4—33no—
172C2×C4×D4—33no—
173C2×#117—33no—
174C2×#117—33no—
175C2×#117—33no—
176C22×C4⋊C4—33no—
177C22×C4⋊C4—33no—
178C22×C4⋊C4—36no—
179C22×C42—33no—
180#389—36no—
181#389—33no—
182#389—33no—
183#389—33no—
184#389—33no—
185#389—33no—
186#389—31no—
187#389—33no—
188#389—33no—
189#389—33no—
190#389—33no—
191#389C2×D313yes—
192#389C2×D313yes—
193#385—36no—
194#385—36no—
195#385—33no—
196#385—33no—
197#385—33no—
198#385—33no—
199#384—33no—
200#384C2×D313yes—
201#382—33no—
202#382—33no—
203#380—36no—
204#380—36no—
205C4×C4⋊C4—33no—
206C4×C4⋊C4—33no—
207C4×C4⋊C4—33no—
208C4×C4⋊C4C2×D313yes—
209C2×#95—36no—
210C23×D3C2413yes—
211C2×C3×Q8C22×C414yes—
212C2×C3×Q8C2×D414yes—
213C2×C3×Q8C2×Q814yes—
214C22×C3×C4C2413yes—
215C22×C3×C4C2411yescharacteristic
216C22×C3×C4C2413yes—
217C22×C3×C4C2411yescharacteristic
218C22×C3×C4C2413yes—
219C22×C3×C4C2411yescharacteristic
220C22×C3×C4C2413yes—
221C22×Dic3C2413yes—
222C22×Dic3C2411yescharacteristic
223C22×Dic3C2413yes—
224C22×Dic3C2411yescharacteristic
225C22×Dic3C2413yes—
226C22×Dic3C2411yescharacteristic
227C22×Dic3C2411yescharacteristic
228#239C4⋊C4164yescomplemented
229C2×D12C22×C4112yes—
230C2×D12C2×D4112yes—
231C2×D12C2×Q8112yes—
232C2×C4×D3—26no—
233C2×C4×D3—26no—
234C2×C4×D3—26no—
235C2×C4×D3—26no—
236C2×C4×D3—212no—
237C2×C4×D3—46no—
238C2×C4×D3—46no—
239C2×C4×D3—43no—
240C2×C4×D3—43no—
241C2×C4×D3—46no—
242C2×C4×D3—46no—
243C2×C4×D3C22×C4124yes—
244C2×C4×D3C22×C4112yes—
245C2×C4×D3C2×D4112yes—
246C2×C4×D3C2×Q8112yes—
247C3×C4⋊C4—26no—
248C3×C4⋊C4—26no—
249C3×C4⋊C4—22no—
250C3×C4⋊C4—46no—
251C3×C4⋊C4—43no—
252C3×C4⋊C4C22×C4112yes—
253C3×C42—26no—
254C3×C42—26no—
255C3×C42C22×C4112yes—
256(C2×D3)⋊C4—26no—
257(C2×D3)⋊C4—26no—
258(C2×D3)⋊C4—212no—
259(C2×D3)⋊C4C22×C4112yes—
260(C2×D3)⋊C4C22×C4112yes—
261C4⋊Dic3—26no—
262C4⋊Dic3—26no—
263C4⋊Dic3—26no—
264C4⋊Dic3—43no—
265C4⋊Dic3C22×C4112yes—
266Dic3⋊C4—22no—
267Dic3⋊C4—22no—
268Dic3⋊C4—26no—
269Dic3⋊C4—26no—
270Dic3⋊C4—46no—
271Dic3⋊C4—43no—
272Dic3⋊C4—43no—
273Dic3⋊C4C22×C414yes—
274Dic3⋊C4C22×C4112yes—
275C4×Dic3—26no—
276C4×Dic3—26no—
277C4×Dic3—22no—
278C4×Dic3—26no—
279C4×Dic3C22×C4112yes—
280C2×#41—38no—
281C2×#41—38no—
282C2×#41—38no—
283C22×Q8C22×D311yescharacteristic
284C22×D4—33no—
285C23×C4—33no—
286C23×C4—36no—
287C23×C4—33no—
288C23×C4—33no—
289C23×C4—36no—
290C4×Q8—24no—
291C4×Q8—24no—
292C4×Q8—38no—
293C4×Q8—64no—
294C4×Q8—64no—
295C4×Q8C4×D318yescomplemented
296C4×D4—324no—
297C4×D4—324no—
298C4×D4—612no—
299C4×D4—612no—
300C4×D4—612no—
301C4×D4—612no—
302#117—324no—
303#117—612no—
304#117—612no—
305C2×C4⋊C4—33no—
306C2×C4⋊C4—33no—
307C2×C4⋊C4—31no—
308C2×C4⋊C4—324no—
309C2×C4⋊C4—33no—
310C2×C4⋊C4—33no—
311C2×C4⋊C4—33no—
312C2×C4⋊C4—33no—
313C2×C4⋊C4—31no—
314C2×C4⋊C4—33no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—36no—
317C2×C4⋊C4—33no—
318C2×C4⋊C4—33no—
319C2×C4⋊C4—33no—
320C2×C4⋊C4—612no—
321C2×C4⋊C4—1212no—
322C2×C4⋊C4C22×D313yes—
323C2×C4⋊C4C22×D313yes—
324C2×C4⋊C4C22×D311yescharacteristic
325C2×C4⋊C4C22×D316yes—
326C2×C4⋊C4C22×D313yes—
327C2×C4⋊C4C22×D313yes—
328C2×C22⋊C4—33no—
329C2×C22⋊C4—33no—
330C2×C22⋊C4—36no—
331C2×C22⋊C4—33no—
332C2×C22⋊C4—33no—
333C2×C42—33no—
334C2×C42—33no—
335C2×C42—33no—
336C2×C42—33no—
337C2×C42—31no—
338C2×C42—612no—
339C2×C42C22×D313yes—
340C2×C42C22×D313yes—
341C2×C42C22×D313yes—
342#95—33no—
343#95—36no—
344#95—36no—
345#95—31no—
346#95—31no—
347#95—33no—
348#95—33no—
349#95—33no—
350#95C22×D313yes—
351#95C22×D313yes—
352C23×C3C2511yescharacteristic
353C22×D3—23no—
354C22×D3—23no—
355C22×D3—23no—
356C22×D3—23no—
357C22×D3C23×C416yes—
358C22×D3C22×D416yes—
359C22×D3C22×Q816yes—
360C3×Q8C2×C4⋊C4116yescomplemented
361C2×C3×C4—21no—
362C2×C3×C4—23no—
363C2×C3×C4—21no—
364C2×C3×C4—23no—
365C2×C3×C4—21no—
366C2×C3×C4—21no—
367C2×C3×C4—23no—
368C2×C3×C4—21no—
369C2×C3×C4—23no—
370C2×C3×C4—23no—
371C2×C3×C4—23no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—23no—
375C2×C3×C4—23no—
376C2×C3×C4C23×C412yes—
377C2×C3×C4C23×C416yes—
378C2×C3×C4C23×C416yes—
379C2×C3×C4C22×D416yes—
380C2×C3×C4C22×Q816yes—
381C2×C3×C42−1+412yes—
382C2×Dic3—21no—
383C2×Dic3—23no—
384C2×Dic3—21no—
385C2×Dic3—21no—
386C2×Dic3—23no—
387C2×Dic3—21no—
388C2×Dic3—21no—
389C2×Dic3—23no—
390C2×Dic3—23no—
391C2×Dic3—23no—
392C2×Dic3—21no—
393C2×Dic3—23no—
394C2×Dic3—23no—
395C2×Dic3—23no—
396C2×Dic3—21no—
397C2×Dic3C23×C412yes—
398C2×Dic3C23×C416yes—
399C2×Dic3C23×C412yes—
400C2×Dic3C22×D412yes—
401C2×Dic3C22×Q812yes—
402C2×Dic3H(5,𝔽2)12yes—
403D12C2×C4⋊C4148yescomplemented
404C4×D3—424no—
405C4×D3—412no—
406C4×D3—412no—
407C4×D3—812no—
408C4×D3C2×C4⋊C4148yescomplemented
409C24—33no—
410#41—364no—
411C2×Q8C2×C4×D314yes—
412C2×Q8D3×D414yes—
413C2×Q8D3×Q814yes—
414C2×D4—312no—
415C2×D4—312no—
416C2×D4—312no—
417C22×C4—31no—
418C22×C4—312no—
419C22×C4—312no—
420C22×C4—312no—
421C22×C4—31no—
422C22×C4—33no—
423C22×C4—31no—
424C22×C4—324no—
425C22×C4—33no—
426C22×C4—31no—
427C22×C4—33no—
428C22×C4—66no—
429C22×C4—66no—
430C22×C4—66no—
431C22×C4—612no—
432C22×C4—66no—
433C22×C4—126no—
434C22×C4—126no—
435C22×C4—123no—
436C22×C4—123no—
437C22×C4—126no—
438C22×C4—126no—
439C22×C4C23×D313yes—
440C22×C4C23×D311yescharacteristic
441C22×C4C23×D313yes—
442C22×C4C23×D311yescharacteristic
443C22×C4C23×D313yes—
444C22×C4C23×D311yescharacteristic
445C22×C4C23×D313yes—
446C4⋊C4—26no—
447C4⋊C4—26no—
448C4⋊C4—22no—
449C4⋊C4—312no—
450C4⋊C4—312no—
451C4⋊C4—34no—
452C4⋊C4—46no—
453C4⋊C4—43no—
454C4⋊C4—66no—
455C4⋊C4—66no—
456C4⋊C4—66no—
457C4⋊C4—62no—
458C4⋊C4—66no—
459C4⋊C4—66no—
460C4⋊C4—62no—
461C4⋊C4—126no—
462C4⋊C4—123no—
463C4⋊C4—123no—
464C4⋊C4—123no—
465C4⋊C4C2×C4×D3112yes—
466C22⋊C4—312no—
467C22⋊C4—312no—
468C22⋊C4—66no—
469C22⋊C4—612no—
470C22⋊C4—66no—
471C42—26no—
472C42—26no—
473C42—312no—
474C42—62no—
475C42—66no—
476C42—66no—
477C42—66no—
478C42C2×C4×D3112yes—
479C22×C3C24×C411yescharacteristic
480C22×C3C23×D411yescharacteristic
481C22×C3C23×Q811yescharacteristic
482C22×C3C2×H(5,𝔽2)11yescharacteristic
483C22×C3C2×2−1+411yescharacteristic
484C22×C3#58511yescharacteristic
485C22×C3#58511yescharacteristic
486C2×D3—26no—
487C2×D3—26no—
488C2×D3—26no—
489C2×D3—26no—
490C2×D3—26no—
491C2×D3—26no—
492C2×D3C22×C4⋊C4112yes—
493C3×C4—26no—
494C3×C4—22no—
495C3×C4—22no—
496C3×C4—43no—
497C3×C4—43no—
498C3×C4—41no—
499C3×C4C22×C4⋊C4112yes—
500Dic3—26no—
501Dic3—22no—
502Dic3—22no—
503Dic3—43no—
504Dic3—43no—
505Dic3—41no—
506Dic3C22×C4⋊C414yes—
507C23—36no—
508C23—36no—
509C23—36no—
510C23—63no—
511C23—63no—
512C23—63no—
513C23—63no—
514C23C24×D311yescharacteristic
515Q8D3×C4⋊C4116yescomplemented
516D4—348no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—23no—
521C2×C4—23no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—23no—
525C2×C4—23no—
526C2×C4—21no—
527C2×C4—23no—
528C2×C4—23no—
529C2×C4—23no—
530C2×C4—23no—
531C2×C4—21no—
532C2×C4—32no—
533C2×C4—32no—
534C2×C4—32no—
535C2×C4—348no—
536C2×C4—32no—
537C2×C4—36no—
538C2×C4—32no—
539C2×C4—61no—
540C2×C4—61no—
541C2×C4—61no—
542C2×C4—61no—
543C2×C4—63no—
544C2×C4—63no—
545C2×C4—61no—
546C2×C4—63no—
547C2×C4—63no—
548C2×C4—61no—
549C2×C4—63no—
550C2×C4—63no—
551C2×C4—63no—
552C2×C4—61no—
553C2×C4—63no—
554C2×C4—1224no—
555C2×C4—1212no—
556C2×C4—1212no—
557C2×C4—2412no—
558C2×C4C22×C4×D312yes—
559C2×C4C22×C4×D316yes—
560C2×C4C22×C4×D316yes—
561C2×C4C2×D3×D416yes—
562C2×C4C2×D3×Q816yes—
563C2×C4#100712yes—
564C2×C3C23×C4⋊C411yescharacteristic
565C2×C3C4×H(5,𝔽2)11yescharacteristic
566C2×C3C4×2−1+411yescharacteristic
567C2×C3#348811yescharacteristic
568C2×C3#348911yescharacteristic
569C2×C3#350311yescharacteristic
570C2×C3#350411yescharacteristic
571D3—212no—
572C22—312no—
573C22—66no—
574C22—66no—
575C22—66no—
576C22—66no—
577C22—66no—
578C22—66no—
579C22C23×C4×D311yescharacteristic
580C22C22×D3×D411yescharacteristic
581C22C22×D3×Q811yescharacteristic
582C22C2×#100711yescharacteristic
583C22#596411yescharacteristic
584C22D3×H(5,𝔽2)11yescharacteristic
585C22#596811yescharacteristic
586C4—22no—
587C4—26no—
588C4—22no—
589C4—34no—
590C4—41no—
591C4—43no—
592C4—43no—
593C4—62no—
594C4—66no—
595C4—62no—
596C4—123no—
597C4—121no—
598C4—123no—
599C4C2×D3×C4⋊C4112yes—
600C3#3425011yescharacteristic, complemented, Sylow
601C2—612no—
602C2C22×D3×C4⋊C411yescharacteristic
603C2C4×#100711yescharacteristic
604C2#8588611yescharacteristic
605C2#8674511yescharacteristic
606C2#8676911yescharacteristic
607C2#8701911yescharacteristic
608C2#8704111yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutatorC22×C3C24×C4
FittingC3×Q8×C4⋊C4C2
socle, cosocleC23×C3C25
prime radicalC22×#239C22
hypercenterQ8×C4⋊C4D3
hypocenterC3#34250
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ Q8×C4⋊C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. C3×Q8×C4⋊C4C2semidirect
5. #82461C2semidirect
6. #82460C2semidirect
7. Q8×Dic3⋊C4C2semidirect
8. #76203C2semidirect
9. #76202C2semidirect
10. #76195C2semidirect
11. D3×#389C2semidirect
12. #74360C2semidirect
13. #74359C2semidirect
14. #73770C2semidirect
15. #73755C2semidirect
25. #5574C4semidirect
76. Q8×C4⋊C4D3semidirect
97. C3×C4×Q8C2×C4semidirect
109. Q8×Dic3C2×C4semidirect
136. #893C2×C4semidirect
140. D3×C4⋊C4C2×C4semidirect
143. C4×D12C2×C4semidirect
145. #872C2×C4semidirect
228. #239C4⋊C4semidirect
295. C4×Q8C4×D3semidirect
360. C3×Q8C2×C4⋊C4semidirect
403. D12C2×C4⋊C4semidirect
408. C4×D3C2×C4⋊C4semidirect
515. Q8D3×C4⋊C4semidirect
600. C3#34250semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#55742
C2×C4×#2392
C3×Q8×C4⋊C42
#824612
#824602
Q8×Dic3⋊C42
#762032
#762022
#761952
D3×#3892
#743602
#743592
#737702
#737552
#342503